Proof of quark confinement and baryon-antibaryon duality: I:
Gauge symmetry breaking in dual 4D fractional quantum Hall superfluidic space-time
A. E. Inopin1 and N. O. Schmidt2
1
Department of Experimental Nuclear Physics, Karazin National University,
Svobody Sq. 4, Kharkiv, 61077, Ukraine (E-mail: [email protected])
2
Department of Mathematics, Boise State University, 1910 University Drive,
Boise, Idaho, 83725, USA (E-mail: [email protected])
(Dated: October 2012)
We prove quark (and antiquark) confinement for a baryon-antibaryon pair and design a welldefined, easy-to-visualize, and simplified mathematical framework for particle and astro physics
based on experimental data. From scratch, we assemble a dual 4D space-time topology and generalized coordinate system for the Schwarzschild metric. Space-time is equipped with “fractional
quantum number order parameter fields” and topological defects for the simultaneous and spontaneous breaking of several symmetries, which are used to construct the baryon wavefunction and its
corresponding antisymmetric tensor. The confined baryon-antibaryon pair is directly connected to
skyrmions with “massive ‘Higgs-like’ scalar amplitude-excitations” and “massless Nambu-Goldstone
pseudo-scalar phase-excitations”. Newton’s second law and Einstein’s relativity are combined to define a Lagrangian with effective potential and effective kinetic. We prove that our theory upgrades
the prediction precision and accuracy of QCD/QED and general relativity, implements 4D versions
of string theory and Witten’s M-theory, and exemplifies M.C. Escher’s duality.
I.
INTRODUCTION
Quarks and antiquarks are the fundamental building
blocks of baryons and antibaryons, respectively. To date,
nature presents an impressive display of mass-energy puzzles in physics, including the creation, annihilation, and
confinement of baryons and antibaryons. The mystery
of quark confinement is a colossal problem in physics;
it is the phenomenon that color charged particles (such
as quarks) cannot be isolated singularly, and therefore
cannot be directly observed1 . In this first paper of the
series, we hunt down this “Great Beast” and prove quarkantiquark confinement for a baryon-antibaryon pair in a
4D space-time, where the dimension of time is circular
rather than linear. We attack the problem from multiple
perspectives simultaneously to establish a well-defined
gauge theory equipped with Legget’s superfluid configuration of Landau’s order parameter fields2 , Laughlin’s
quasiparticles3 , a baryon wavefunction, antisymmetric
tensors, and more. From scratch, we construct a topological solution and Lagrangian that intertwines Newtonian4
and Einsteinian concepts5 , improves the predictive capability of quantum chromo-dynamics (QCD) and quantum
electro-dynamics (QED)6 , reduces dimensional complexity of string theory and M-theory7 , exhibits M.C. Escher’s duality8 , and is directly supported by a diverse
experimental array. In short, the dual baryon and antibaryon quantum states are encoded with quantum number order parameters of fractional statistics for quasiparticles with baryon wavefunction antisymmetry. We
prove quark-antiquark confinement in terms of Laughlin excitations3 that dynamically arise due to our fractional quantum Hall superfluidic (FQHS) space-time and
topology inspired by the quasiparticle interferometer experiments of Goldman9 . We prove that the quarks and
antiquarks confined to the holographic ring “cancel out”
due to the CPT-Theorem. Spontaneous symmetry breaking (SSB) generates massless “Nambu-Goldstone pseudoscalar phase-excitations” 10–13 and massive “Higgs-like
scalar amplitude-excitations” 14 of Laughlin statistics3 .
First, we provide conceptual quark confinement proof in
Section II. Second, we provide mathematical quark confinement proof in Sections III, IV, V, and VI.
In Section II, we prepare for our quark confinement proof by conceptually aligning the reader to
our 4D FQHS space-time scenario. We investigate
the two dynamical scales that arise in the doubleconfinement, double-stereographic gravitational superlensing, and double-horizons inherent to baryons and
antibaryons. We prove that a baryon-antibaryon pair
is composed of three distinct quark-antiquark pairs,
which form three corresponding “thin color-electric
flux tubes” 15 of Laughlin excitations3 and fractional
statistics16 . We discuss the hadronization process and
the modified Gribov QED/QCD vacuum, where all properties in 3D Schwarzschild space can be inferred from
the analogue of the 2D gauge field on the six-coloring
kagome lattice manifold. Additionally, we venture to the
interaction between the boson propagators and gravity
by introducing “gravitational birefringence”.
In Section III, we begin mathematically constructing
the topology and framework for our quark confinement
proof in FQHS. We discuss the surface and generalized
Riemann coordinates used to encode our FQHS spacetime. We extend the definition of complex numbers and
use them to represent locations on the 1D Riemann surface; we prove that the complex numbers are both scalars
and Euclidean vectors. We define axis constraints for
the vectors, which let us construct a powerful 2D generalized coordinate system on the surface equipped with
the Pythagorean identity; the locations may always be
expressed in terms of right triangles with real and imag-
The Hadronic Journal • Volume 35 • Number 5 • October 2012
inary components.
II.
In Section IV, we explore the three distinct topological sub-surface zones for a baryon and antibaryon using
set and group theory. We formally define the zones using trichotomy and our generalized coordinates for 2D
and 3D space. We prove that the time-like region is a
holographic ring—a closed curve and simple contour of
points, which can be scaled to, for example, the Fermi
radius. We formally define space and time as being dual.
Additionally, we demonstrate that the time-like region
represents the U (1) and SU (2) symmetry groups, which
is isomorphic to the SO(3) orthogonal group; all 3D properties are inferred directly from the 2D holographic ring
for the SU (2) gauged Bose-Einstein condensate.
ALIGNMENT TO CONFINEMENT:
CONCEPTUAL PROOF
At the Fermi scale, a baryon’s event horizon confinement radius baryon = 2Mbaryon strongly depends on it’s
mass Mbaryon , which can vary in size in accordance to
its quark composition identified by the Standard Model.
This is known as baryon confinement (for three-quark
confinement) and is modeled as a baryon bag. Similarly, an antibaryon’s event horizon confinement radius
antibaryon = 2Mantibaryon strongly depends on it’s mass
Mantibaryon . This is known as antibaryon confinement
(for three-antiquark confinement) and is modeled as an
antibaryon bag. A baryon and its antibaryon merge to reflect three-pair quark-antiquark confinement. For example in a proton-antiproton pair, an antiproton of antimass
Mantibaryon = Mantiproton = 1 GeV precisely counterbalances a proton of mass Mbaryon = Mproton = 1 GeV due
to antiferromagnetic ordering and the CPT-Theorem.
On this scale we identify the general mechanism, namely
Baryon-Antibaryon Confinement (BAC), which is responsible for the dynamics. It is based on the appearance
of a critical radius 2M = baryon = antibaryon for quarkantiquark confinement at the 1 Fermi scale and the appropriate generalized dynamics—effective gravito-strong
interaction. So in gravity plus electromagnetism, there
is one interesting mechanism—radiation trapping just on
the horizon’s surface, that is a coherent particle accumulation structure18 of fractional statistics and toroidal vortex 20 . The toroidal vortex, that stores information as in
the holographic hypothesis15 , intertwines the baryon and
antibaryon confinement mechanisms, creating BAC. The
toroidal vortex forms between the spherical shells defined
at the inner confinement radius 2M and the outer confinement radius 3M = 3M (based on the effective potential); 2M and 3M correspond to the “horizon” and “imaginary surface”, respectively, in Figure 6 of Witten15 ; there
are two distinct quantum critical points imposed by an
antibaryon or baryon for the double-stereographic gravitational superlense with the meta-material, acoustic,
double-negative refractive index, and sub-wavelength features of21–24 —see Figure 1. These facts are evident from
the DIS modeling results of the hadronization process25 .
Quark-hadron duality in jet formation in DIS leads to
a two-step process of hadronization, with two scales appearing: large Q20 Λ2QCD and small Q20 ∼ 1GeV 2 .
An alternative approach in DIS, namely “Local Parton
Hadron Duality”, also leads to the two dynamical scales:
k⊥ = Q0 ∼ ΛQCD and k⊥ = Q0 ∼ 1 GeV25 . Both models of the hadronization process give us the numbers in
accord with our model 2M ∼ 0.2 − 0.3 fm and 3M ∼ 1
fm. Another fresh perspective can be taken from the
“Glue drops” model26 , where the authors gave firm evidence of the existence of a non-perturbative scale, smaller
1
than the usual ΛQCD
∼ 1 fm, which is related to gluonic
degrees of freedom. The evidence for the presence of a
semi-hard scale in hadronic structure is reviewed from
many venues. The most notable effects are: QCD sum
In Section V, we define the Baryon Wavefunction
(BWF) of fractional quantum number order parameters (OP) for our quark (q) and antiquark (q̄) confinement proof. Additionally, we discuss the amplitudeexcitations14 and phase-excitations10–13 for Laughlin
quasiparticles3 experienced by the BWF OPs in our
FQHS space-time scenario. For this, we express the full
BWF antisymmetries and CPT-transformations.
In Section VI, we express the Lagrangian in terms of effective potential and effective kinetic for our FQHS spacetime scenario. For this, we apply both Newtonian and
Einsteinian concepts to the q and q̄ confinement proof
and thereby incorporate effective force, effective mass,
and effective acceleration.
In Section VII, we prepare a concise correspondence
to the authors of the Yuan-Mo-Wang (YMW) baryonantibaryon SU (3) model17 . In doing so, we relay the
importance of the YMW model and report on a number
of similarities between it and our confinement scenario.
Additionally, we contrast the models by identifying key
distinctions and suggest that our model may upgrade the
YMW model’s state space and accuracy, and thereby extend its prediction horizon. Ultimately, we realize that
both constructions exhibit remarkable similarities, and
that it may be possible in the near future to consolidate
these ideas into a single framework.
In Section VIII, we conclude our paper with a brief
recapitulation and outlook where we suggest future research trajectories.
To summarize, in this first paper of the series we introduce the topologies, vacuum, generalized coordinates,
fractional statistics, quantum number OPs, BWF, gauge
symmetry breaking, and Lagrangian for the q and q̄ confinement proof in 4D FQHS space-time; for the scenario,
we provide a series of colorful depictions and an array
of experiments supporting this construction. In the next
paper(s) of this series, we will extend our confinement
scenario by discussing the anyons, phase locking18 , Hubius helix (HH)19 , attractive and repulsive gravitational
effects of quasiparticle signals on the Lagrangian, modified Gullstrand–Painlevé reference frames, and Magnification Effect.
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The Hadronic Journal • Volume 35 • Number 5 • October 2012
Rashba spin-orbit coupling29 —see Figure 4. The quarks
(and leptons) are “split” into three distinct excitation degrees of freedom, namely spinon, holon, and orbitons3,30 ;
the Laughlin excitations of the FQHS 3-branes obey fractional statistics; luminal quasiparticle signals of the RSZ
“sea” execute a closed path around the NSZ “island” of
superluminal quasiparticle signals and thus acquire statistical phase9 —see Figure 5.
In QCD, BAC is a difficult strong coupling problem,
but a somewhat similar phenomenon in nature is much
better understood in QED. The Meissner effect is the
fundamental observation that a superconductor expels
magnetic flux. Suppose that magnetic monopoles become available for study and that we insert a monopoleantimonopole pair into a superconductor, where the two
poles are separated by a large distance x. What will happen? A monopole is inescapably a source of the magnetic
flux, but magnetic flux is expelled from a superconductor.
So the optimal solution to this problem, energetically, is
that a thin, non-superconducting tube forms between the
monopole and the antimonopole. The magnetic flux is
confined to this region, which is known as an AbrikosovGorkov flux tube (or a Nielsen-Olesen flux tube in the
context of relativistic field theory). The flux tube has
a certain nonzero energy per unit length, so the energy
required to separate the monopole and antimonopole by
a distance x grows linearly in x, for large x.
As a non-Abelian gauge theory, QCD has fields rather
similar to ordinary electric and magnetic fields but obey
a nonlinear version of Maxwell’s equations. Quarks and
antiquarks are particles that carry the QCD analog of
electric charge and are confined in our QCD vacuum
just as ordinary magnetic charges would be in a superconductor. The color-electric-magnetic quark monopoles
and anticolor-electric-magnetic antiquark antimonopoles
may be separated by a large distance x to form nonAbelian dipoles: red-antired, green-antigreen, and blueantiblue “thin color-electric flux tubes” 15 . Now from the
Aharonov–Casher (AC) effect and Aharonov–Bohm (AB)
effect duality31–33 , it is evident that this analogy immediately leads to the idea that the QCD vacuum is to a
superconductor, just as electricity is to magnetism, and
just as the AC effect is to the AB effect—see Figure 3.
In34 , the author considered a relativistic string model,
where a massless quark moves at the speed-of-light in a
circular orbit. One can see clearly the x = x0 = 2M coordinate represents an event horizon or “impenetrable barrier” and the quark moves in the “half harmonic oscillator” potential. When combined with the phenomenological aspects of35,36 , a strong QCD/QED string model for
the q q̄ pairs with the associated quasiparticles3 emerges
in our scenario. So for the q q̄ pairs we identify both
open-ended (“linear”) fermionic strings and the closed
(“non-linear”/circular) bosonic strings vibrating in our
conjugate and dual space-time. All of this is supported
by Glue drops26 , where the energy of a QCD string is
concentrated in a thin color-electric flux tube15 of radius 2M = 0.3 fm. All such particles and quasiparticles
rules gives 0.3 fm radius of the corresponding form factor, lattice gives 0.2-0.3 fm for the correlation length,
instanton radius peaks approximately at 0.3 fm, diffractive gluon bremsstrahlung in hadronic collisions leads to
k⊥ for the gluons in a proton of about 0.7 GeV27 . At
higher scales, chiral symmetry breaking is restored and
the vacuum does not feel apparent existence of quark and
gluon condensates, which spoil the chiral symmetry from
the start—the mechanism for the spontaneous breaking
of chiral symmetry and spontaneously emergent behavior
of chaos theory on the Lagrangian.
All together, this brings us to the concept of BaryonAntibaryon Duality (BAD), which is responsible for the
stereographic superlensing23 dynamics. At rest, the
massless red, green, and blue quarks are confined to a
baryon and circulate counter-clockwise along it’s event
horizon as a left-handed HH19 at the speed of light to generate effective mass, such that all observable baryons are
white; the visible colored quarks are non-Abelian colorelectric-magnetic monopoles 28 which emit red, green, and
blue light-rays to render a baryon. Similarly, the resting antired, antigreen, and antiblue antiquarks are confined to a antibaryon and circulate clockwise along it’s
event horizon as a right-handed HH19 to generate effective antimass, such that all “observable” antibaryons are
black (antiwhite); the “visible” anticolored antiquarks are
non-Abelian anticolor-electric-magnetic antimonopoles 28
which emit antired, antigreen, and antiblue light-rays
to render an antibaryon; the relative direction of circulation (with corresponding winding number) distinguishes between mass (i.e. Mproton ) and antimass (i.e.
Mantiproton ). For BAD, the baryon and antibaryon bags
are dual, opposite, reverse, and inverse, and are therefore modeled as a Baryon-Antibaryon Bag (BAB). The
quark and antiquark trajectories follow Wilson loops
and form a self-consistent10 SU (2) gauged Bose-Einstein
condensate29 . The electro-strong duality of the potentials continuously transform in FQHS space-time in accordance with 1D, 2D, and 3D skyrmions29 .
This rich concept of duality enables us to compute observables in time-like regions, given the physics in spacelike regions, and vice-versa. Upon considering these dual
fields, the idea of two distance scales comes up naturally.
Our 1D Riemmann surface (2D holographic information
structure) is divided into three distinct topological subsurfaces for quasiparticles:
1. Non-Relativistic Space Zone (NSZ) or “Micro” distance scale of superluminal signals,
2. Time Zone (TZ), and
3. Relativistic Space Zone (RSZ) or “Macro” distance
scale of luminal signals.
The Riemannian holographic ring unit circle represents
the TZ and is isometrically embedded on the surface;
it bifurcates 3D space to establish the NSZ, such that
0 < x < 2M , and the RSZ, where 2M < x < ∞—recall
Figure 1. The gauge field is a 3D analogue of the TZ’s
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The Hadronic Journal • Volume 35 • Number 5 • October 2012
FIG. 1. The Riemannian holographic ring unit circle of two counter-propagating edge channels defines the TZ for quarkantiquark confinement and is isometrically embedded on the 1D Riemann surface. The toroidal vortex between two dynamical
scales for a double-stereographic gravitational superlense: the spherical shells located at critical radius 2M = 2M and 3M =
3M .
no longer coincide with the geometrical light-cones fixed
by the local Lorentz invariance of space-time, but depend explicitly on the local curvature. This formulation
agrees with the von Karman flow and symmetry breaking
of37 , the kaleidoscope of exotic quantum phases in the 2D
frustrated model of38 , and the deviant Fermi liquid of39 ,
where the TZ serves a Bose metal as in40 . All this works
in 4D space-time.
The q q̄ pairs for a baryon-antibaryon pair are “superbound” to the vacuum41 as coupled oscillators42 (see
Figure 2) and form red-antired, green-antigreen, and
blue-antiblue Nambu-Goldstone pions, which are NambuGoldstone bosons; the SSB of the three distinct pions generates colored amplitude-excitations14 and phaseexcitations10–13 . The q q̄ pairs of the three distinct thin
color-electric flux tubes are confined to the TZ, which is
a Riemannian holographic ring unit circle on a 1D Riemann surface equipped with a six-coloring (three coloring plus three anticoloring) kagome lattice manifold generalization of43 with antiferromagnetic ordering3 . The
ring exhibits the Rashba and fractional quantum Hall
effects44 , along with spin-Hall current45 and chiral magnetic moments46 . The q q̄ pairs are uniformly arranged
along the kagome lattice with the triangular chirality of47
and the self-assembling observables of18,48 (recall Figure
3). The quasiparticles of the SU (2) gauged Bose-Einstein
condensate are direct 3D analogs of the spontaneously
emerging QED and QCD. The kagome lattice hexagonal structure is self-similar to, for example, graphene,
which explains the “plasmaron” observations in quasifreestanding doped graphene49 and the “soundaron” ob-
on the Riemann surface which generate effective mass
(and antimass) are projected along the “z-axis” to effective 3D space (recall Figure 4). Here, events are represented on the Lagrangian using generalized coordinates
in Schwarzschild space-time on the Riemann surface.
Viewed in certain classes of inertial frames, a superluminal signal travels backwards in time. In QED, Feynman diagrams involve a virtual e+e- pair that influences
the photon propagator. Here, positrons are replaced with
electron-holes. This gives a photon an effective mass
(or antimass) on the order of the Compton wavelength
for the electron (or electron-hole); leptons are split into
quasiparticles3,30 . All of this is generalized to QCD,
where a virtual q q̄ pair influences the gauge boson propagator in FQHS space-time; the propagator is a function which returns a probability amplitude of 1 for the
quarks and baryon confined to the TZ. In both QED
and QCD, if the space-time curvature has a comparable scale, then an effective boson-gravity interaction is
induced; the Higgs-like amplitude excitations14 for the
baryon-antibaryon pair impose effective mass for baryons
and quarks, and effective antimass for antibaryons and
antiquarks. This depends explicitly on the curvature, in
violation of the Strong Equivalence Principle. The boson
velocity is changed and light-ray no longer follows the
shortest possible path—it bifurcates to both the NSZ
and RSZ distance scales. Moreover, if the space-time
is anisotropic, this change can depend on the boson’s
polarization as well as direction. This is the quantum
phenomena of “gravitational birefringence”. The effective
light-cones for boson propagation in gravitational fields
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The Hadronic Journal • Volume 35 • Number 5 • October 2012
relations of65 , Fermi liquid deviations of63 , non-linear optics, analogue gravity, and photon emissions analogous to
the Hawking radiation as in66 , and Andreev reflections
of67,68 ; the TZ’s current continuously undergoes chargetransformation between the NSZ’s and RSZ’s supercurrent. The q q̄ resonances form the exotic meson and broad
locking states as in69 . The q q̄ pairs and their waves are
phase locked, spontaneously aligning to form dynamical
1D coherent accumulation structures with time-periodic
flows18 and a von Kármán vortex street20 with impact
parameters.
servations of50 . The quarks can also be thought as moving along the “caustics” inside the toroidal vortex, where
the quark’s trajectories are trapped between the dual
scale dynamics—they are “gliding” along the surface and
are reflected back to the center. The dual confinement
boundaries located at 2M and 3M act as reflecting and
focusing stereographic superlenses. So baryons and antibaryons become seashells closed on 3M 51 .
When we come to the vacuum estate, the richness of BAD is shining brightly: Gribov’s QED/QCD
vacuum41 resembles a complicated structure of UnruhBoulware-Hartle-Hawking’s black hole vacuum and is fed
with solid-state physics along with notions of forbidden zones, Fermi surfaces, particles and holes to encode the BAB on the Riemann surface. But there
are some new diagrams that arise with the new zones,
and novel types of excitations—enabling us to upgrade
Gribov’s model. This new vacuum differs drastically
from Dirac’s vacuum and contains a total of 18 zones
for the six-coloring (kagome lattice) manifold on the
Riemann surface—Figure 4; these zones are populated
with quasiparticles3,30 spontaneously generated by the
q q̄ pairs confined to the TZ with the spin-orbit coupling
of45,53–55 . The TZ acquires a geometric phase31,32,56 ,
so the quasiparticles confined to the TZ are dual to
those signals propagated across the NSZ and RSZ zones.
Laughlin’s fractional quantization16 is axiomatic in this
scenario. At proper temperature and pressure, the vacuum is consistent with Chernodub57 . Clearly, in treating
the BAD and superlensing dynamics, it is very convenient to separate the RSZ and NSZ degrees of freedom
(Born–Oppenheimer approximation).
The NSZ and RSZ both represent superconductive,
FQHS 3-branes interconnected by the TZ, which serves
as a common (2D) surface boundary at 2M . The baryon
and antibaryon are spinning objects confined to the TZ so
they generate whirlpools on both 3-brane distance scales
in accordance with seashells closed on 3M 51 , thereby
exhibiting the Magnus effect58 and generating a vortexantivortex dance52 ; these whirlpools are described on the
Riemann surface using spirals (i.e. weighted Fibonacci
sequence and/or golden spiral). The TZ is a topological Mott insulator for30,53,59,60 , a Fermi surface as in61 ,
a Goldman-Laughlin quasiparticle interferometer of two
counter-propagating edge channels as in9 , a Gedanken
interferometer as in62 , a quantum critical point as in3,63 ,
and a non-perturbative, self-consistent, SU (2) gauged
Bose-Einstein condensate as in10 that satisfies Novikov’s
self-consistency principle as in64 ; a picture emerges of the
vacuum as a conductor instead of “Dirac’s insulator”, with
a new mass scale that reflects the position of the “Fermi
surface” 41 . The six-coloring antiferromagnetic alignment
of the q q̄ pairs spontaneously generate the physical behavior of the strong interaction as in3 and thereby triggers parity doubling, CPT violations, and different polarization rotation velocities on both the NSZ and RSZ
distance scales simultaneously. Here, we identify the
Dirac quantization and spin-charge magnetic monopole
III. THE SPACE-TIME SURFACE AND
GENERALIZED RIEMANN COORDINATES
Let X be the 1D Riemann surface. We define the
complex number x = xR + xI as a position-point and
position-vector on X; x ∈ X is both a complex scalar
and Euclidean vector with amplitude |x| and phase hxi,
which are analogous to magnitude and direction in conventional notation. The orthogonal components of x,
namely xR ∈ R1 and xI ∈ I1 as axis-constrained real
and imaginary Euclidean vectors, respectively (where in
this case I denotes imaginary rather than irrational); the
simple trichotomy axis-constraints for the R-axis are
xR > 0 ⇔ hxR i = 2π = 0,
xR = 0 ⇔ hxR i = @,
xR < 0 ⇔ hxR i = π,
(1)
(2)
(3)
and for the I-axis are
π
,
2
xI = 0 ⇔ hxI i = @,
3π
,
xI < 0 ⇔ hxI i =
2
xI > 0 ⇔ hxI i =
(4)
(5)
(6)
such that
|xR | = |x| cos(hxi),
|xI | = |x| sin(hxi),
(7)
(8)
with Pythagorean form
|x|2 = x2R + x2I , ∀x ∈ X.
(9)
Thus, we’ve defined the 2D generalized (Riemann) coordinate system of X as
2D
X : (x) = (xR + xI ) = (xR , xI ) = (|x|, hxi), ∀x ∈ X,
(10)
with respect to the unique reference origin-point O ∈ X,
such that (O) = (0 + 0i) = (0, 0i) = (0, 0π); (x) =
(xR + xI ) are 1D Riemann coordinates, (xR , xI ) are 2D
Cartesian coordinates, and (|x|, hxi) are Polar coordinates; a Complex-Cartesian-Polar synchronized and generalized coordinate system. The real and imaginary axisconstraints ensure that the generalized coordinates may
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The Hadronic Journal • Volume 35 • Number 5 • October 2012
FIG. 2. Schematic of the multiple synchronized quark and antiquark solid-state oscillators (colored and anticolored circles)
coupled to generate frequencies for the SU (2) gauged Bose-Einstein condensate with skyrmions29 in the loop configuration
based on the work of Afshari42 ; the coupling circuits (gray triangles) shift the phase of the oscillators.
FIG. 3. The loop-induced zero-energy dynamics are described as “gluon dynamics”. The 3 distinct q q̄ pairs for the baryonantibaryon pair are “superbound” as coupled oscillators42 to the Fermi surface in the upgraded Gribov vacuum generalized from41
and are confined to the kagome lattice antiferromagnet on the six-coloring manifold. The q q̄ pairs spontaneously generate
phase-excitations (massless and pseudo-scalar)10–13 and “Higgs-like” amplitude-excitations (massive and scalar)14 Laughlin
excitations3 . The toroidal vortex along the Riemannian holographic ring unit circle for a baryon and/or antibaryon is defined
as a toroidal vortex between the spherical shells located at critical radius 2M = 2M and 3M = 3M ; double stereographic
superlenses23 for two dynamical scales27 . An affinity exists between BAD and M.C. Escher’s duality, where the combined
baryon event horizon and antibaryon event horizon at 2M exhibit the double horizon phenomena8 . The q q̄ pairs confined to
the TZ form thin color-electric flux tubes15 in the QCD vacuum of the NSZ and exhibit the AC effect, while thin magnetic
flux tubes in the RSZ superconductive region exhibit antiferromagnetic ordering and the AB effect; the QCD vacuum is to a
superconductor, just as electricity is to magnetism, and just as the AC effect is to the AB effect. This model exhibits vortexantivortex dancing52 and confirms the spontaneous appearance of a stable 3D skyrmion in the SU (2) gauged Bose-Einstein
condensate of29 confined to the Riemannian holographic ring unit circle on our 1D Riemann surface.
always be expressed as a right-triangle with Pythagorean
properties.
So how to we extend our 2D generalized coordinates
of Definition (10) to 3D Schwarzschild space? Well, for
a baryon or antibaryon of scale M (located precisely at
the origin position-point O ∈ X) we define the 3D generalized (Schwarzschild) coordinate system of X as
3D
X : (ux , |x|, hxi) = (
M
, |x|, hxi), ∀x ∈ X.
|x|
IV.
ZONES
We define T as the TZ of X. So T is a topological representation of a Riemannian unit circle, where
the critical radius of T is scaled and normalized to precisely 2M = 2M = π2 scalar . We prove BAC on T .
scalar is the time unit scale-normalizing constant and
2M is the inner confinement radius of T . Next, we
define the circumference and wavelength of T , namely
Tλ = Tcircumf erence = Twavelength = 2πscalar , as being
(11)
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The Hadronic Journal • Volume 35 • Number 5 • October 2012
FIG. 4. The gauge-invariant TZ delineates the NSZ and RSZ: a 2-sphere which is dual to both 3-branes, where the SU (2)
Bose–Einstein condensate and gauge field is a 3D analogue of the Rashba spin-orbit coupling of the TZ, supporting the 1D,
2D, and 3D Skyrmion structures29 (all). The baryon-antibaryon pair comprises the three distinct q q̄ pairs and is modeled as a
BAB in the new Gribov vacuum with 18 quasiparticle signal zones (bottom).
where we define all T position-points as time-points and
equivalent to the (normalized) Mikhail Grimov’s area filling conjecture 70 : Tarea = Tλ ; T ⊂ X is a closed curve
and simple contour of surface position-points.
We use zone trichotomy to simultaneously define the
TZ and SZ regions of X: we define X− and X+ as the
NSZ and RSZ of X, respectively. The surface T delineates the topological sub-surfaces X− and X+ on X; T is
a Mott insulator30 and Fermi surface41 which delineates
two dual superconductors30,53,59,60,67,68 . Thus, ∀x ∈ X
we know that precisely one of the following conditions
must be satisfied:
|x| < 2M ⇔ x ∈ X− ,
|x| = 2M ⇔ x ∈ T,
|x| > 2M ⇔ x ∈ X+ ,
X− = {s ∈ X : |s| < 2M },
X+ = {s ∈ X : |s| > 2M },
where we define all S = X− ∪X+ position-points as spacepoints. So clearly,
22M = |t|2 = |tR |2 + |tI |2 , ∀t ∈ T,
2
2
2
|x| = |xR | + |xI | , ∀x ∈ X.
(18)
(19)
So T is isometrically embedded in X with the one-to-one
holographic mappings f : T ,→ X and f : T → X− ∪ X+
with dual simultaneous bijections
(12)
(13)
(14)
fT ime : X− ←- T ,→ X+ ,
fSpace : X− ,→ T ←- X+ ,
where clearly X− ∩ T = T ∩ X+ = X− ∩ X+ = ∅ and
X− ∪ T ∪ X+ = X. Hence, T is the multiplicative group
of all non-zero complex 1-vectors, such that
T = {t ∈ X : |t| = 2M },
(16)
(17)
(20)
(21)
for our dual space-time; we’ve proven that T is dual to
X− and T is also dual to X+ . Interestingly, this formulation may provide a simplification to the Riemann-Hilbert
(15)
7
The Hadronic Journal • Volume 35 • Number 5 • October 2012
FIG. 5. The upgraded Gribov QCD/QED vacuum with 18 zones for quasiparticle signals pertaining to a BAB on the 1D
Riemann surface. The q q̄ pairs are confined to the TZ, which is dual to the NSZ and the RSZ. The six-coloring spinon,
holon, and orbiton excitations are spontaneously generated and confined to the TZ, which acquires a geometric phase; the TZ
excitations are dual to those of the NSZ and RSZ 3-branes.
problem as expressed in, for example,71 . Now because T
is a type of Riemannian circle and holographic ring, we
know it is a 2-sphere for the SU (2) gauged Bose-Einstein
condensate29 . Thus, for the position-point and positionvector t ∈ T we apply Definition (10) to express the
2-sphere generalized and synchronized 2D Riemann coordinates
road to the t’Hooft and Maldacena holographic model
for high-energy physics—all the 3D properties are inferred directly from the 2D (Riemannian holographic
ring) domain72 .
We define T as a “fermiwire,” which is nothing
more than a “nanowire” 55,73 with Rashba spin-orbit
coupling30,32,44 on the Fermi scale. The spin geometric phase for electrons in32 is applied directly to the spin
Hall effect44 , effective spin-dependent flux, and Andreev
reflections67,68 of the quarks and antiquarks confined to
the universal curve T (the holographic ring with uniform
radius |t| = 2M ) embedded in X; the duality derivation between the AAS effectrand the AC effect of32 is
2
Φ
2mt hti|t|
written for T as Φmag
⇐⇒
1
+
, ∀t ∈ T,
2
~
0 /2
2D
T : (t) = (tR +tI ) = (tR , tI ) = (|t|, hti) = (, hti), ∀t ∈ T,
(22)
and in 3D Schwarzschild coordinates
3D
T : (ut , |t|, hti) = (
M
, |t|, hti), ∀t ∈ T.
|t|
(23)
Now because ∀t ∈ T we have the uniform radius |t| =
2M , we can alternatively drop the |t| amplitude coordinate and just use the hti phase coordinate to directly specify position-points on the 1D non-linear surface. Therefore, T is
where Φmag is the magnetic flux, Φ0 = h/e is the one flux
quantum period, hti = α is the amplitude and strength of
the Rashba spin-orbit interaction, and mt is the effective
mass; the left term is the AAS effect flux and the right
term is the time-reversal AC effect oscillation unit with
effective spin-dependent flux for the conductance modulation and voltage dependence observations of the AAS
amplitude at zero magnetic field32 . This formulation is
crucial to our six-coloring quark-antiquark configuration
for the BAC scenario because the magneto-resistance oscillations of32,74 along T are attributed to the AAS effect.
• the 1D circular Abelian group U (1);
• the 2D spherical non-Abelian group SU (2); and
• isomorphic to the 3D orthogonal non-Abelian group
SO(3),
which directly supports 1D, 2D, and 3D skyrmions29 . So
parity doubling27 is synonymous of the term degeneracy, and Escher gave an example of how one can establish 2D - 3D correspondence8 . We see here again the
8
The Hadronic Journal • Volume 35 • Number 5 • October 2012
FIG. 6. The TZ is dual to both distance scales and imposes the double-confinement and double-lensing of M.C. Escher’s
duality8 ; it is a stereographic superlense23 between the two 3-brane distance scales.
FIG. 7. Inopin’s interpretation of M.C. Escher’s double-horizons of8 is directly connected to the q → q̄ transitions, past-present
switching, time-reversal operation, and CPT-Theorem on the Riemann surface: time is circular and non-linear, so the past is
the future. The quarks switch back and forth between the conjugate space-time regions with the appearance and disappearance
of 3 quantum critical points in the QCD phase diagram.
the BWF using OPs and Laughlin statistics3 in our nonAbelian SU (2) gauge theory. In the theory of superfluidity the OP measures the existence of Bose condensed
particles (Cooper pairs) and is given by the probability
amplitude of such particles. The inter-particle forces be-
V. THE WAVEFUNCTION DEFINITION OF
FRACTIONAL QUANTUM NUMBER ORDER
PARAMETERS
Landau introduced the concept of OPs2 , which we define as complex scalar fields10 on X. Here, we construct
9
The Hadronic Journal • Volume 35 • Number 5 • October 2012
tween quarks and antiquarks, and between 4 He and between 3 He atoms, are rotationally invariant in spin and
orbital space and, of course, conserve quantum number27 .
The latter symmetry gives rise to gauge symmetry, which
is broken in any superfluid. First, for the theory of
isotropic superfluids like a BCS superconductor or superfluid 4 He, we define the global OP ψ = ψR + ψI as a
complex number (which inherits the notation similar to
x as defined in Section III without loss of generality); ψ
is both a complex scalar and Euclidean vector with the
amplitude |ψ|14 and phase hψi components10 . Then for
local gauge SSB, we define the OP ψ[x] as the complex
scalar field
ψ[x] = ψ[x]R + ψ[x]I , ∀x ∈ X,
orbital rotation symmetry, as in liquid crystals 27 . Including the gauge symmetry, three symmetries are therefore
broken in superfluid 3 He. The theoretical discovery that
several simultaneously broken symmetries can appear in
condensed matter was made by Antony Leggett, and represented a breakthrough in the theory of anisotropic superfluids, 3 He2 . This leads to superfluid phases whose
properties cannot be understood by simply adding the
properties of systems in which each symmetry is broken
individually. Such phases may have long range order in
combined, rather than individual degrees of freedom. So
to construct a strong BWF constraint for BAC to T , we
“Cooper pair” the OP set of strongly conserved quantum
numbers
(24)
ΦOP = {ψC , ψI , ψJ },
where |ψ[x]| and hψ[x]i are the “gauge” amplitude and
phase components local to x ∈ X, respectively, in accordance to Englert10 . Furthermore, we define ∆|ψ[x]|
and ∆hψ[x]i as the change of the OP’s amplitude and
phase due to a “massive Higgs-like amplitude-excitation”
and “massless Nambu-Goldstone phase-excitation” components, respectively—see Figures 8 and 9. Since the
Mott insulator and stereographic superlense T is dual to
both X− and X+ , we express Equation (24) specifically
for time-points as the parametric function
ψ(t) = ψ(t)R + ψ(t)I , ∀t ∈ T,
(26)
which is listed in Table I; the spin-orbit coupling of44,45,65
applies directly to T , where ψJ (t) is identical to the
“BSO -vector” of73 , such that
ψJ (t) = ψS (t) + ψL (t), ∀t ∈ T.
(27)
The q q̄ pairs confined to T on the six-coloring kagome lattice manifold are located at position-points r, g, b, r̄, ḡ, c̄ ∈
T ; they adhere to the uniformly-arranged position-point
constraints
(25)
where the SU (2) gauge-invariant T acquires a Berry–
Aharonov–Anandan geometric phase as in56 ; T is an
ordered medium equipped with an OP space for topological defects. The classical energy density distribution
along T is a function of the OP ψ(t); within the ordered
(superfluid) phase, Nambu-Goldstone and Higgs modes
arise from the hψ(t)i and |ψ(t)|, respectively, where the
energy density transforms into a function for T with a
minimum at |ψ(t)| = 014 . So |ψ(t)| is excited with a
periodic modulation of the spin-orbit coupling, which
amounts to a “shaking” of the energy density (effective) potential for topological deformations along T in
accordance with14 . Furthermore, because the baryonantibaryon pair is confined to T on the kagome lattice of antiferromagnetic ordering43 , we define the BWF
for the six-coloring position-points {r, g, b} subsetT and
{r̄, ḡ, b̄} ⊂ T of three colored quarks and three anticolored antiquarks in the vacuum, respectively (recall Figures 3 and 4).
Above the critical temperature the system is invariant
under an arbitrary change of the phase hψ[x]i → hψ[x]i0 ,
i.e. under a gauge transformation. Below the critical
temperature a particular value of hψi is spontaneously
preferred. In anisotropic superfluids, additional symmetries can be spontaneously broken, corresponding to multiple OP components of the BWF. In 3 He—the best studied example with multiple OP components—the pairs are
in a spin-triplet state, meaning that rotational symmetry in spin space is broken, just as in a magnet. At the
same time, the anisotropy of the Cooper-pair wavefunction in orbital space calls for a spontaneous breakdown of
hri = hr̄i ± π, hgi = hḡi ± π, and hbi = hb̄i ± π, (28)
with uniform amplitudes |r| = |g| = |b| = |r̄| = |ḡ| =
|b̄| = 2M , and antiferromagnetic ordering
hψJ (r)i = hψJ (r̄)i ± π,
hψJ (g)i = hψJ (ḡ)i ± π, and
(29)
(30)
hψJ (b)i = hψJ (b̄)i ± π,
(31)
(recall Figure 3). A little flight of imagination lead us to
this new approach, where the OPs ∀t ∈ T are “Cooper
paired” to form a Leggett superfluid B phase of2 with
azimuthal “alpha” phase angle hti; the OPs ∀ψ ∈ ΦOP
rotate freely in 2D and 3D space, while the superfluid B
phase angle hti ∈ {hri, hgi, hbi, hr̄i, hḡi, hb̄i} between them
remains constant. Such phases form correlated helices
along T , serving as constraints for the BWF—see Figure
10.
Next, we construct our BWF for the BAB states. For
a baryon and antibaryon centered on the origin-point
O ∈ X and confined to T we define the full baryon and
antibaryon states as
Ψtotal (r, g, b) = Ψ(r) × Ψ(g) × Ψ(b) and
(32)
Ψtotal (r̄, ḡ, b̄) = Ψ(r̄) × Ψ(ḡ) × Ψ(b̄),
(33)
respectively, for the BAC and BAD; the red, green, and
blue quark wavefunctions respectively located at timepoints r, g, b ∈ T on the three-coloring triangular sub10
The Hadronic Journal • Volume 35 • Number 5 • October 2012
FIG. 8. A complex scalar field ψ(t) experiences a massive “Higgs-like” amplitude-excitation 14 (right), which is characteristic of
the Nambu-Goldstone scalar boson SSB order parameter fluctuations discussed by10 ; a classical wave imposes volume effects
and stretches the vacuum field.
FIG. 9. A complex scalar field ψ(t) experiences a phase-excitation (right), which is characteristic of the Nambu-Goldstone
pseudo-scalar SSB order parameter fluctuations discussed by10 ; a classical wave imposes rotational effects on the vacuum field
in accordance with vacuum degeneracy.
the BWF for the three distinct q q̄ pairs that are confined
to T along the six-coloring kagome lattice manifold (recall
Figure 3). So the antisymmetric BWF is described with
the six-coloring components
lattice are
def
Ψ(r) = ψC (r) × ψJ (r) × ψI (r) × r, Ψ(r) = hr|Ψi,
(34)
def
Ψ(g) = ψC (g) × ψJ (g) × ψI (g) × g, Ψ(g) = hg|Ψi,
(35)
def
Ψ(b) = ψC (b) × ψJ (b) × ψI (b) × b, Ψ(b) = hb|Ψi,
(36)
Ψ(r, r̄) = −Ψ(r̄, r),
Ψ(g, ḡ) = −Ψ(ḡ, g), and
(40)
(41)
Ψ(b, b̄) = −Ψ(b̄, b),
(42)
for the confined quark and antiquark (two-particle) cases.
So for Definition (32) and the related six-coloring Definitions (31–37), we define the full BWF antisymmetrization via the covariant antisymmetric metric tensor: the
2D antisymmetric BWF matrix
!
0
Ψtotal (r, g, b)
(43)
Ψtotal (r̄, ḡ, b̄)
0
and the antired, antigreen, and antiblue antiquark wavefunctions respectively located at time-points r̄, ḡ, b̄ ∈ T
on the three-anticoloring triangular sub-lattice are
def
Ψ(r̄) = ψC (r̄) × ψJ (r̄) × ψI (r̄) × r̄, Ψ(r̄) = hr̄|Ψi,
(37)
def
Ψ(ḡ) = ψC (ḡ) × ψJ (ḡ) × ψI (ḡ) × ḡ, Ψ(ḡ) = hḡ|Ψi,
(38)
and the expanded 3D antisymmetric BWF matrix


0 Ψ(r) Ψ(g)


(44)
Ψ(r̄) 0 Ψ(b) 
Ψ(ḡ) Ψ(b̄) 0
def
Ψ(b̄) = ψC (b̄) × ψJ (b̄) × ψI (b̄) × b̄, Ψ(b̄) = hb̄|Ψi;
(39)
11
The Hadronic Journal • Volume 35 • Number 5 • October 2012
TABLE I. The quantum number order parameters for the BWF states on the 1D Riemann surface X. Here, ψJ = ψS + ψL
and ψJ (t) = ψS (t) + ψL (t) for the spin-orbit coupling of the holographic confinement ring T ⊂ X.
Order Parameter
Color Charge
Isospin
Orbital Angular Momentum
Spin Angular Momentum
Total Angular Momentum
Symbol
C
I
L
S
J
Global
ψC = ψCR + ψCI
ψI = ψIR + ψII
ψL = ψLR + ψLI
ψS = ψSR + ψSI
ψJ = ψJR + ψJI
FIG. 10. Leggett’s2 six distinct superfluid B phase angles for the three
lattice of antiferromagnetic ordering3,43 . The superfluid B phase angles
the OPs as they rotate freely in 2D and 3D space; this long range order
along T ; this concept serves as a strong BWF constraint and applies to
ψJ (t) and ψI (t) are shown, but ψC (t) is also correlated with hti.
q q̄ pairs confined to T along the six-coloring kagome
hri, hgi, hbi, hr̄i, hḡi, hb̄i remain constant and correlate
applies ∀t ∈ T , ∀ψ ∈ ΦOP , to form correlated helices
all OPs for a given time-point. In this diagram, only
the OP charge transformation(s), ∀ψ ∈ ΦOP ,
for T . So given complex tangent vectors µ and ν we
define
gx (µ, ν) = −gx (ν, µ) ∈ C, ∀x ∈ X;
Local
ψC [x] = ψC [x]R + ψC [x]I
ψI [x] = ψI [x]R + ψI [x]I
ψL [x] = ψL [x]R + ψL [x]I
ψS [x] = ψS [x]R + ψS [x]I
ψJ [x] = ψJ [x]R + ψJ [x]I
(45)

ψ(t) ! 7→ −ψ(t), !
the tensor describes the X curvature (“vector phase”)




hgx (µ, ν)i and the field strength (“vector amplitude”)

−ψ(t)R
 ψ(t)R
7→
,
|gx (µ, ν)| at a position-point x ∈ X. The Levi-Civita
ψ(t)I
−ψ(t)I
C:
(47)
!
!
symbol for the color singlet function is




|ψ(t)|
|ψ(t)|



7→
,


hψ(t)i
hψ(t)i ± π
+1 if (r, g, b) is (1, 2, 3), (2, 3, 1), or (3, 1, 2)
ζrgb = ζ rgb =
0 if r = g or g = b or b = r


−1 if (r, g, b) is (3, 2, 1), (2, 1, 3), or (1, 3, 2)
(46)
The CPT-Theorem is a fundamental property of T .
the parity transformation(s) (in generalized 2D Riemann
Hence, for a baryon or antibaryon of scale M we have
coordinates for 3D Schwarzschild space) is the flip in the
12
The Hadronic Journal • Volume 35 • Number 5 • October 2012
sign of the one coordinate


 
−tR
tR





 


7  −tI  ,
 tI  →



 M
−M
 |t| 
P :  |t| 

|t|
|t|




 



7 hti ± π  ,
hti →


 M
−M
|t|
|t|
and time reversal transformation(s)


t ! 7→ −t, !




−tR
 tR
7→
,
tI
−tI
T :
!
!




|t|
|t|


7→
,

hti ± π
hti
Einstein’s Fx , the EP per unit of particle effective mass
mx , is defined as
v
"
#
u
J 2
(
)
EP [x] u
m
x
Fx = mx ax =
, ∀x ∈ X,
= t(1 − 2ux ) 1 +
mx
|x|2
(48)
(53)
where the ax along coordinate phase hxi is
ax =
where kx is the wave vector, ε(kx ) is the dispersion relation, and ex is the point charge in an external electric
field E.
(49)
VII.
A BRIEF CORRESPONDENCE TO YUAN,
MO, AND WANG
So now that we’ve presented our model, we realize
that it inherits exceptional components from the YMW
baryon-antibaryon SU (3) model17 ; in particular, it is evident that their ideas and applied methodology strongly
support our BAC proof. Here, we compare and contrast
our model with the YMW model and provide a brief correspondence aimed at unifying the conceptual and mathematical components of both schemes. We’ve identified
the YMW model as a well-defined framework that provides crucial insight into the nature of (theoretical and
experimental) particle physics. Moreover, we’ve found
that the construction of our BAC proof has inevitably
led us to assemble a theory and framework that effectively replicates core expressions of the YMW paradigm,
bringing us along similar paths of exploration.
To summarize, the YMW model is a nonet scheme
which predicts many new baryon-antibaryon bound
states and their possible productions in quarkonium decays and B decays17 . It is designed to classify the
increased number of experimentally observed enhancements near the baryon-antibaryon threshold. It is largely
based on the Fermi-Yang-Sakata (FYS) model76,77 , in
which mesons were interpreted as baryon-antibaryon
bound states17 . The discovery of the increased number
of baryon-antibaryon enhancements near thresholds reminds the YMW authors of the era prior to the development of the SU (3) quark model over half a century ago,
when the so-called elementary particles emerged one-byone17 . So YMW return to the FYS model and extend
various aspects of it.
First, we identify a few key similarities between our
model and the YMW model17 . In short, both schemes:
VI. THE LAGRANGIAN: EFFECTIVE
POTENTIAL AND EFFECTIVE KINETIC
Here, we express the gauged SSB in our FQHS spacetime scenario on X, which is applicable to both 2D and
3D space; the Lagrangian is defined as
(50)
using our generalized coordinates, where EK [x] and
EP [x] are the effective kinetic and effective potential, respectively, for a position-point x. From75 the gauge boson’s EP is defined as
√
1 − 2ux
, ∀x ∈ X.
(51)
EP [x] =
|x|
The EP depends on the Schwarzschild geometry but not
on the choice of orbit. Only one EP is required to analyze the motion of all radiation (including radio waves,
radar pulses, gamma rays, etc.). It is important to stress
EP differences and similarities between a massive particle and its massless limit: radiation-rays. Next, the EK
is defined as
1
1 Fx 2
mx vx2 =
v , ∀x ∈ X,
2
2 ax x
(54)
x
which comprise a CPT-transformation, ∀t ∈ T . We see
that for Definitions (47), (48), and (49) there are multiple equivalent transformations for each case because
the generalized Riemann coordinates of Definition (10)
and the OP Definition (24) use synchronized ComplexCartesian-Polar values (where the magnitude and direction of the Polar components are replaced with amplitude
and phase, respectively).
L[x] = EK [x] − EP [x], ∀x ∈ X,
1 X ∂ 2 ε(kx )
ex Emx , ∀x ∈ X,
~2 m ∂khxi ∂kmx
(52)
• Share the core hexagon structure equipped with a
baryon-antibaryon pair.
where Fx is the effective force, where mx is the effective
mass, ax is the effective acceleration, and vx is the effective velocity of the particle at x in the FQHS space-time.
• Intertwine a hexagon and circle(s) to encode certain
fundamental aspects of the baryon-antibaryon state
space.
EK [x] =
13
The Hadronic Journal • Volume 35 • Number 5 • October 2012
• The YMW model does not define a Lagrangian
that is consistent with Newtonian and Einsteinian
paradigms in 4D space-time, whereas ours does;
but the YMW model can be simplified and formulated as such to highlight these additional spacetime and dynamical system relationships.
• Predict many new baryon-antibaryon bound states.
• Accommodate the enhancements near the baryonantibaryon mass thresholds.
• Predict the increase and decrease of masses for the
dual three-quark system (the baryon and the antibaryon) with respect the baryon-antibaryon mass
threshold.
• The circular structures of the YMW model do
not consider a well-defined circular dimension of
time that is modeled as a Riemannian holographic ring unit circle, stereographic gravitational
superlense23 , Gedanken interferometer62 , and Mott
insulator30 that is dual to both superconducting 3-branes which simultaneously triggers CPTviolations on the dual distance scales, whereas ours
model does employ these notions; but the YMW
model can be equipped with this paradigm so
it is consistent with, for example, M.C. Escher’s
duality8 .
• Organize quark-antiquark pairs into flavorless
mesons.
• Account for pseudo-scalar meson states.
• Support the possible charmonium and B decay
modes as listed by YMW.
Second, we identify a few key dissimilarities between
our model and the YMW model17 . In doing so, we clarify that our model is not only consistent with the YMW
model, but upgrades many of its components by simplifying the nonet representation from SU (3) to SU (2) and
extending its state space, accuracy, and predictive capability. In short, the schemes differ in that:
• The YMW model is not based upon a dual spacetime topology equipped with topological deformations of intertwined superluminal (non-relativistic)
signals and luminal (relativistic) signals, whereas
our model does incorporate this arrangement; but
the YMW model can be equipped with such energy
deformations, signal classifications, and topological
structure.
• The hexagon structure of the YMW model does not
incorporate the six-coloring kagome lattice antiferromagnet, whereas our model does; but the YMW
model can be equipped with this powerful lattice
structure to fundamentally enhance its representational capability.
At this point we’ve provided a brief report that compares and contrasts some high-level aspects between the
two models. For the future, this suggests that we may
clarify a plethora of new ideas and fundamental relationships through an objective consolidation of both frameworks.
• The YMW model does not model the quarks as
coupled oscillators which generate effective mass,
whereas our model does; but the YMW model can
be equipped with quark coupled oscillators to attribute its mass increases and decreases to the effective mass generated by the interconnected oscillators.
VIII.
• The YMW model employs classic quantum numbers
(for total angular momentum, isospin, strangeness,
and charge) to encode the system state, whereas
our model employs quantum number OPs (for total angular momentum, isospin, and color charge)
to encode the system state with additional precision; but the YMW model can be equipped with
quantum number OPs of fractional statistics to
construct a well-defined (baryon and antibaryon)
wavefunction in state space, where the OPs are correlated with superfluid B phases2 .
CONCLUSION AND OUTLOOK
In this first paper of the series, we introduced the
topologies, vacuum, generalized coordinates, fractional
statistics, BWF quantum number OPs, gauge symmetry breaking, and Lagrangian for our BAC proof and
BAD in 4D FQHS space-time that complies with Newtonian and Einsteinian mechanics, and low-dimensional
implementations of string theory and M-theory. In the
next paper(s) of this series, we will extend our quarkantiquark confinement scenario by discussing the anyons,
phase locking, HH, attractive and repulsive gravitational
effects of quasiparticle signals on the Lagrangian, modified Gullstrand–Painlevé reference frames, and Magnification Effect.
In our opinion, this proof of quark-antiquark confinement with the amplitude–excitations and phase–
excitations begins to reveal additional fundamental
mechanisms and relationships inherent to our universe.
In doing so, we’ve been able to shed more light on a
number of mysterious concepts in nature, including antibaryons, baryons, baryon asymmetry, creation, annihilation, double horizons, and FQHS space-time. We
• The YMW model only accounts for pseudoscalar mesons, whereas our model accounts
for pseudo-scalar and scalar mesons, while directly associating these components to massless Nambu-Goldstone phase-excitations11–13 and
massive Higgs-like amplitude-excitations14 , respectively; but the YMW model can be equipped with
scalar mesons, phase-excitations, and amplitudeexcitations that correspond to a “Goldstone Family” of gauge bosons for SSB.
14
The Hadronic Journal • Volume 35 • Number 5 • October 2012
suspect that these formulations, which are inspired by
a plethora of experimental data, can be used to construct a unified field theory in the near future, thereby
advancing physics to the “next level.” Through global co-
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operation, competition, hard work, and creativity, these
powerful concepts can be further scrutinized, extended,
and applied to virtually all areas of mathematics, science,
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